In the following, we illustrate the statistical analysis of microdeformation processes for the example of compression experiments conducted on Mo micropillars. We focus exclusively on the data analysis aspects; for a description of the experimental details, the reader is referred to the original articles [1, 2].

The deformation curves of compressed micropillars are characterised by a strongly intermittent behaviour: deformation proceeds as a discrete sequence of “deformation events” during which the plastic deformation rate increases significantly, and these events are separated by intervals of near-elastic stress increase. The shape of the corresponding stress-strain curves depends on the loading mode. On the one hand, in displacement-controlled deformation, rapid plastic flow during a deformation event leads to elastic unloading. Hence, the stress-strain curves assume a serrated shape where each deformation event corresponds to a stress drop. In stress-controlled loading, on the other hand, rapid deformation leads to a strain increase at almost constant stress. The corresponding, almost-horizontal parts of the stress-strain curve are separated by much steeper intervals of low plastic activity where the stress increases in a nearly elastic manner. Hence, the stress-strain curves assume a staircase-like shape, as seen in Figure 1. Both stress and strain increments scatter widely, leading to a variation in flow stresses that increases with increasing strain.

An obvious first step towards a statistical characterisation of deformation curves consists in determining the average value and the statistical variation of flow stresses as functions of strain and specimen size. This is shown in Figures 2 and 3. It can be seen that both the flow stress and the flow stress variation of Mo micropillars are size dependent. They rise rapidly at the onset of deformation and saturate above strains of about 10%.

Figure 2 Average deformation curves of Mo micropillars and nanopillars of different diameters. Each curve represents an average of eight samples. Data from [1, 2].

Figure 3 Scatter of deformation curves of Mo micropillars and nanopillars of different diameters. Each curve represents the standard deviation of the flow stresses of eight samples. Data from [1, 2].

Figure 4 demonstrates that the size-dependent flow stress and flow stress variation scale in approximate proportion with each other. Both dependences can be well described by inverse power laws, <*σ*>∼<Δ*σ*^{2}>^{1/2}∼*d*^{-a}, where *a≈*0.5 and *d* is the micropillar diameter.

Figure 4 Size dependence of the flow stress and flow stress variation of [100]-oriented Mo single crystals (both parameters determined at 15% strain).

For a more detailed statistical analysis, we consider the sequence of strain bursts and stress increases observed in stress-controlled compression tests. To analyse the statistical properties of this sequence, we start out from time records of stress, plastic strain, and strain rate. After filtering out external sources of noise in the signals, we construct a sequence of deformation events by imposing a threshold on the plastic strain rate (for details, see [2]). The ensuing series of stress and strain jumps can be statistically characterised in terms of probability distributions of stress increments Δ*σ* and strain increments Δ*ε*. The strain bursts exhibit a scale-free power law distribution:

with an exponent *κ* close to 3/2 as also reported for many other materials [20]. The distributions do not depend appreciably on system size (Figure 5). By contrast, the distributions of stress increments show a clear system size dependence (Figure 6). They are not scale-free – in fact, they can be represented well by Weibull distributions,

Figure 5 Probability distribution of strain increments determined in stress-controlled compression of [100]-oriented Mo micropillars of different sizes. The straight line indicates a power law distribution with exponent 3/2.

Figure 6 Cumulative probability distributions of stress increments determined in stress-controlled compression of [100]-oriented Mo micropillars of different sizes. The straight line is a Weibull distribution with parameter 0.87. Inset, Mean stress increment as a function of sample diameter.

with a shape parameter (Weibull exponent) *β* close to 1, i.e., we find near-exponential behaviour. This parameter does not depend on the system size. The mean stress increment (and also the stress parameter Δ*σ*_{0} of the Weibull distributions), by contrast, is inversely proportional to the system size (inset in Figure 6). This poses the question how the size dependences of the flow stress and of the stress increment distribution relate to each other.

Before we address this question, a word of caution is needed. It is in the nature of power-law distributions with exponents between 1 and 2 that p(Δ*ε*) contains a very large number of small jumps as the probability density diverges at small Δ*ε*. This “infrared divergence” must be mitigated by introducing a cut-off Δ*ε*_{min} at small sizes. The actual value of Δ*ε*_{min} is at first glance of little importance, as a simple calculation demonstrates that, for small Δ*ε*_{min}, the (infinitely many) small events that are cut out produce only a negligible fraction of the overall strain. However, the value of the cut-off in the strain increment distribution is of crucial importance for the distribution of stress increments: it is evident that the mean stress increment simply equals the mean flow stress divided by the number of events that have occurred up to that stress. Reducing Δ*ε*_{min} (the smallest strain increment still counted as a strain jump) increases the number of events and hence reduces the mean value of the stress increment distribution. Thus, any discussion of the size dependence of the stress increment statistics is meaningless unless we specify how we define Δ*ε*_{min}.

In our analysis of Mo micropillar deformation, the smallest event size was implicitly defined to be inversely proportional to the system size (the imposed requirement was that the corresponding displacement increment had to exceed the typical level of machine-induced oscillations in the displacement record). Using the relation Δ*ε*_{min}∼*d*^{-1}, we may now try to understand the relation between the size dependence of the flow stress and the stress increment statistics.

To this end, we formulate a minimal statistical model by assuming that the stress and strain increments in a sequence are mutually uncorrelated random variables. In addition, we assume that the sequences of stress and strain increments constitute two independent, stationary stochastic processes such that the statistics of deformation curves is completely characterised by the respective probability distributions. Under these assumptions, the following statistical arguments can be put forward:

The total number of events is equal to the total strain divided by the average strain increment:

For a power law distribution of exponent 3/2, the average is the arithmetic mean of the upper and lower cut-offs of the power law scaling regime. Hence, the average strain increment is

The upper cut-off of the scaling regime is given by the total strain (at strain *ε*, no event with Δ*ε*>*ε* can possibly have occurred):

Δ*ε*_{max}∼*ε.*

The average stress increment is given by the flow stress (determined at strain *γ*) divided by the number of strain increments that have occurred up to that strain:

The analysis of flow stresses for the Mo experiments indicates that *σ*∼*d*^{-1/2}. Furthermore, Δ*ε*_{min}∼*d*^{-1} due to the manner the stress-strain curves were analysed [2]. Inserting both relations into Eq. (5), we find that <Δ*σ*>∼*d*^{-1} as observed. Thus, the observed size dependences of the stress and stress increment statistics are mutually consistent within the framework of our minimal statistical model.

Temporal intermittency of microscale plasticity goes along with spatial localisation as the sequence of distinct deformation events corresponds to a pattern of localised slip bands. Using surface analysis methods, Schwerdtfeger et al. [22, 23] demonstrated that the local strains in slip bands obey the same statistics as the temporal strain increments during strain bursts, supporting the conjecture that each strain burst corresponds to the formation of a localised slip band. Although the slip band heights were demonstrated to exhibit scale-free statistics, the slip band spacings were exponentially distributed, supporting the idea that slip band nucleation can be envisaged as an uncorrelated Poisson process. Using these ideas, a statistical model of the formation and evolution of slip band patterns was proposed in [23], which produced a good agreement with the overall surface morphology observed in experiments on alkali halide single crystals.

Unfortunately, despite these successes, it is difficult to build statistical models of microplasticity exclusively upon experimental data. For evident reasons, experimentators shun the boring task of preparing and testing large numbers of identical specimens. Even in a carefully conducted and well-documented series of experiments, it is rare to find statistics based upon more than a dozen specimens for each set of physical and geometrical parameters. Moreover, it is difficult to fully characterise and reproduce – let alone independently control – the statistical parameters of the initial dislocation microstructure in different samples. As a consequence of incomplete characterisation and poor statistics, there is little agreement as to the physical origin of fluctuations and size effects. Even for a seemingly straightforward parameter as the size effect exponent *a*, different leading groups keep advocating different values (compare, e.g., the reviews [4, 6]). To overcome the difficulties related to poor statistics and insufficient information regarding microstructures, it is useful to resort to discrete dislocation dynamics simulations. Although such simulations have problems of their own (restriction to small samples and small strains), they allow to independently control the initial dislocation microstructure and simulate reasonably large ensembles by unsophisticated “farming out” of computations over multiple computers.

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